A subset C of edges in a k-uniform hypergraph H is a loose Hamilton cycle if C covers all the vertices of H and there exists a cyclic ordering of these vertices such that the edges in C are segments of that order and such that every two consecutive edges share exactly one vertex. The binomial random k-uniform hypergraph H
k
n,p
has vertex set [n] and an edge set E obtained by adding each k-tuple e ∈ (
$\binom{[n]}{k}$
) to E with probability p, independently at random.
Here we consider the problem of finding edge-disjoint loose Hamilton cycles covering all but o(|E|) edges, referred to as the packing problem. While it is known that the threshold probability of the appearance of a loose Hamilton cycle in H
k
n,p
is
$p=\Theta\biggl(\frac{\log n}{n^{k-1}}\biggr),$
the best known bounds for the packing problem are around
p = polylog(
n)/
n. Here we make substantial progress and prove the following asymptotically (up to a polylog(
n) factor) best possible result: for
p ≥ log
C
n/
n
k−1, a random
k-uniform hypergraph
H
k
n,p
with high probability contains
$N:=(1-o(1))\frac{\binom{n}{k}p}{n/(k-1)}$
edge-disjoint loose Hamilton cycles.
Our proof utilizes and modifies the idea of ‘online sprinkling’ recently introduced by Vu and the first author.